Blind Sensor Calibration using Approximate Message Passing

The ubiquity of approximately sparse data has led a variety of com- munities to great interest in compressed sensing algorithms. Although these are very successful and well understood for linear measurements with additive noise, applying them on real data can be problematic if imperfect sensing devices introduce deviations from this ideal signal ac- quisition process, caused by sensor decalibration or failure. We propose a message passing algorithm called calibration approximate message passing (Cal-AMP) that can treat a variety of such sensor-induced imperfections. In addition to deriving the general form of the algorithm, we numerically investigate two particular settings. In the first, a fraction of the sensors is faulty, giving readings unrelated to the signal. In the second, sensors are decalibrated and each one introduces a different multiplicative gain to the measures. Cal-AMP shares the scalability of approximate message passing, allowing to treat big sized instances of these problems, and ex- perimentally exhibits a phase transition between domains of success and failure.Comment: 27 pages, 9 figure

Compressed Optical Imaging

We address the resolution of inverse problems where visual data must be recovered from incomplete information optically acquired in the spatial domain. The optical acquisition models that are involved share a common mathematical structure consisting of a linear operator followed by optional pointwise nonlinearities. The linear operator generally includes lowpass filtering effects and, in some cases, downsampling. Both tend to make the problems ill-posed. Our general resolution strategy is to rely on variational principles, which allows for a tight control on the objective or perceptual quality of the reconstructed data. The three related problems that we investigate and propose to solve are 1. The reconstruction of images from sparse samples. Following a non-ideal acquisition framework, the measurements take the form of spatial-domain samples whose locations are specified a priori. The reconstruction algorithm that we propose is linked to PDE flows with tensor-valued diffusivities. We demonstrate through several experiments that our approach preserves finer visual features than standard interpolation techniques do, especially at very low sampling rates. 2. The reconstruction of images from binary measurements. The acquisition model that we consider relies on optical principles and fits in a compressed-sensing framework. We develop a reconstruction algorithm that allows us to recover grayscale images from the available binary data. It substantially improves upon the state of the art in terms of quality and computational performance. Our overall approach is physically relevant; moreover, it can handle large amounts of data efficiently. 3. The reconstruction of phase and amplitude profiles from single digital holographic acquisitions. Unlike conventional approaches that are based on demodulation, our iterative reconstruction method is able to accurately recover the original object from a single downsampled intensity hologram, as shown in simulated and real measurement settings. It also consistently outperforms the state of the art in terms of signal-to-noise ratio and with respect to the size of the field of view. The common goal of the proposed reconstruction methods is to yield an accurate estimate of the original data from all available measurements. In accordance with the forward model, they are typically capable of handling samples that are sparse in the spatial domain and/or distorted due to pointwise nonlinear effects, as demonstrated in our experiments

Autocalibrated Signal Reconstruction From Linear Measurements Using Adaptive Gamp

In this paper, we reconstruct signals from underdetermined linear measurements where the componentwise gains of the measurement system are unknown a priori. The reconstruction is performed through an adaptation of the message-passing algorithm called adaptive GAMP that enables joint gain calibration and signal estimation. To evaluate our approach, we apply it to the problem of sparse recovery and compare it against an l(1)-based approach. We numerically show that adaptive GAMP yields excellent results even for a moderate amount of data. It approaches the performance of oracle GAMP where the gains are perfectly known asymptotically